Time: 3 hours                                                                             Maximum marks: 80

General Instructions:

1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions.

2. Section A has 18 MCQ’s and 02 Assertion-Reason based questions of 1 mark each.

3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each.

4. Section C has 6 Short Answer (SA)-type questions of 3 marks each.

5. Section D has 4 Long Answer (LA)-type questions of 5 marks each.

6. Section E has 3 source based/case based/passage based/integrated units of assessment of 4 marks each with sub-parts.

 

Section –A

(Multiple Choice Questions)

Question 1:

The values of x and y for which the matrices 

            

are equal        

(a) 7,

(b) 7,

(c) 7, 7

(d) No value of x and y

 

Question 2:

The value of  is

(a)  *  

(b)  *

(c)  *  

(d) None of these

 

Question 3:

The values of x, y and z where the vectors a = xi + 2j + zk and b = 2i + yj + k are equal 

(a) 2, 2, 1

(b) 2, 1, 2

(c) 1, 2, 2

(d) 2, 1, 1

 

Question 4:

The principal value of sin^-1  is

(a)  

(b)

(c)

(d)

 

Question 5:

The distance of the plane 2x – 3y + 6z + 14 = 0 from origin, is

(a) 2

(b) 3

(c) 4

(d) 5

 

Question 6:

If P(A) = 0.8, P(B) = 0.5 and P  = 0.4, then the value of P  is

(a) 0.32

(b) 0.48

(c) 0.56

(d) 0.64

 

Question 7:

If a and b are the order and degree of differential equation y  + x³  + xy = cos x, then

(a) a < b

(b) a = b

(c) a > b

(d) not defined

 

Question 8:

If a = 2i - j + 2k and b = -i + j - k then the unite vector in the direction of (a + b) is

(a)

(b)

(c)

(d)

 

Question 9:

The value of  is

(a)

(b)

(c)

(d)

 

Question 10:

If a function f : R -> R defined by f(x) = |x|, x ∈ R, then the function is

(a) One-One

(b) Many-one

(c) One-One and Many-one

(d) Neither One-One nor Many-one

 

Question 11:

A 2 * 2 matrix whose elements are given by a_ij = 2i – j, is

 

 

Question 12:

The points of discontinuity of f, where f is defined by

f(x) =             if x < 0

             -1         if x ≥ 0, is/are

(a) All positive real numbers only

(b) All negative real numbers only

(c) All real numbers

(d) There is no point of discontinuity

 

Question 13:

The equation of motion of an aeroplane are x = 5t, y = -10t, z = 5t where t is given in seconds and distance measured is in km then path of aeroplane is

(a) Straight line

(b) Parabola

(c) Hyperbola

(d) Circle

 

Question 14:

If A = 4i + 3j and B = 3i + 4j then |A| + |B| is

(a) 5

(b) 5

(c) 10

(d) 10

 

Question 15:

The sum of cofactors of all elements of   is

(a) 2

(b) 4

(c) -5

(d) -3

 

Question 16:

If the solution of a differential equation  represents a circle then the value of a is

(a) 2

(b) -2

(c) 3

(d) -4

 

Question 17:

Rekha and Aman appeared of an interview for two vacencies. The probability of Rekha’s selection is  and for Aman is . The probability that both of them are rejected, is

(a)

(b)

(c)

(d)

 

Question 18:

Feasible region of a LPP has shown in the given figure.

The corner points of feasible region are

(a) (60, 0), (120, 0), (60, 30), (40, 20)

(b) (0, 0), (0, 60), (120, 0), (40, 20)

(c) (0, 0), (60, 0), (120, 0), (60, 30), (40, 20)

(d) (60, 0), (0, 60), (120, 30), (40, 20)

 

ASSERTION-REASON BASED QUESTIONS

In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R).

Choose the correct answer out of the following choices.

(a) Both (A) and (R) are true and (R) is the correct explanation of (A).

(b) Both (A) and (R) are true but (R) is not the correct explanation of (A).

(c) (A) is true but (R) is false.

(d) (A) is false but (R) is true.

 

Question 19:

Let W be the set of words in the English dictionary. A relation R is defined on W as R = {(x, y) ∈ W * W such that x and y have at least one letter is common}

ASSERTION (A): R is reflexive.

REASON (R): R is symmetric.

 

Question 20:

 

 

Section – B

Question 21:

If  = θ, then what is the value of cos θ?

 

Question 22:

Find the approximate change in the volume V of a cube of side x metres caused by increasing side by 1%.

 

Question 23:

Find the equation of the normal to curve y² = 4x + 5 at the point (-1, 1).

 

Question 24:

Evaluate:    

 

Question 25:

The radius of a circle is increasing at the rate of 0.7 . What is the rate of increase of its circumference?

 

Section – C

Question 26:

Find the value of

 

Question 27:

A die is thrown. If E is the event ‘the number appearing is a multiple of 3’ and F be the event ‘the number appearing is even’ then find whether E and F are independent?

 

Question 28:

Evaluate:

 

Question 29:

Verify that the function  is a solution of the differential equation

 

 

Question 30:

Solve the following LPP.

Maximize Z = 7.5x + 5y  

subject to the constraints,

2x + y ≤ 60       

x ≤ 20               

2x + 3y ≤ 120  

x, y ≥ 0             

 

Question 31:

A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is .

 

Section – D

Question 32:

Find the area of the region bounded by the curve y² = 4x and the line x = 3.

 

Question 33:

Show that the relation R defined in the set A of all triangles as R = {(T₁, T₂): T₁ is similar to T₂}, is equivalence relation. Consider three right angle triangles T₁ with sides 3, 4, 5, T₂ with sides 5, 12, 13 and T₃ with sides 6, 8, 10. Which triangles among T₁, T₂ and T₃ are related?

 

Question 34:

By using properties of determinants, show that:

 

 

Question 35:

The scalar product of the vector i + j + k with a unit vector along the sum of vectors 2i + 4j – 5k and λi + 2j + 3k is equal to one. Find the value of λ.

 

Section – E

Question 36:

Suppose a dealer in rural area wishes to purpose a number of sewing machines. He has only Rs 5760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18.

Based on the above information given, answer the following questions.

(i) Let x and y denotes the number of electronic sewing machines and manually operated sewing machines purchased by the dealer. If it is assume that the dealer purchased at least one of the given machines, then

(a) x + y ≥ 0               (b) x + y < 0               (c) x + y > 0                   (d) x + y ≤ 0

(ii) Let the constraints in the given problem is represented by the following inequalities

x + y ≤ 0

360x + 240y ≤ 5760

x, y ≥ 0

Then which of the following point lie in its feasible region.

(a) (0, 24)                  (b) (8, 12)                   (c) (20, 2)                   (d) None of these

(iii) If the objective function of the given problem is maximize Z = 22x + 18y, then its optimal value occur at

(a) (0, 0)                     (b) (16, 0)                      (c) (8, 12)                        (d) (0, 20)

 

Question 37:

The Government declare that the farmers can get Rs 300 per quintal for their onion on 1^st July and after that the price will be dropped by Rs 3 per quintal per extra day. Ramu’s brother has 80 quintal of onions in the field on 1^st July and he estimates that the crop is increasing at the rate of 1 quintal per day.

Based on the above information given, answer the following questions.

(i) If x is the number of days after 1^st July, then the price and quantity of onion respectively can be expressed as

(a) Rs (300 – 3x), (80 + x) quintals          (b) Rs (300 – 3x), (80 - x) quintals                      (c) Rs (300 + 3x), (80 + x) quintals          (d) Rs (300 + 3x), (80 - x) quintals

(ii) Revenue R as a function of x can be represented as

(a) R(x) = 3x² – 60x - 24000                     (b) R(x) = -3x² + 60x + 24000

(c) R(x) = 3x² + 40x - 16000                     (d) R(x) = 3x² – 60x - 14000

(ii) On which day should Ramu’s father harvest the onions to maximize his revenue?

(a) 11^th July              (b) 20^th July                  (c) 12^th July                     (d) 22^nd July

 

Question 38:

A company produces three products every day. Their Production on certain day is 45 tons. It is found that the production of third exceeds the production of first production by 8 tons while the total production of first and third product is twice the production of second product.

Using the concept of matrices and determinants, answer the following questions.

(i) If x, y and z denotes the quantity (in tons) of first, second and third product produced respectively, then which of the following is true.

(a) x + y + z = 45           (b) x + 8 = z             (c) x - 2y + z = 0           (d) All of above

(ii) x : y : z is equal to

(a) 12 : 13 : 20              (b) 11 : 15 : 19       (c) 15 : 19 : 11              (d) 13 : 12 : 20

(ii) Which of the following is not true?

(a) |A| = |A’|                     

(b)  =                                       

(c) A is skew symmetric matrix of odd order, then |A| = 0                         

(d) |AB| = |A| + |B|

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